Determining action selection policies of an execution device

ABSTRACT

Disclosed herein are methods, systems, and apparatus for generating an action selection policy (ASP) for an execution device. One method includes obtaining an ASP in a current iteration; obtaining a respective first reward for each action in a current state; computing a first reward of the current state based on the respective first rewards for the actions and the ASP; computing a respective regret value of each action based on a difference between the respective first reward for the action and the first reward for the current state; computing an incremental ASP based on the respective regret value of the each action in the current iteration; computing a second reward for the current state based on the incremental ASP; determining an ASP in a next iteration based on the second reward of the current state; and controlling actions of the execution device according to the ASP.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of PCT Application No. PCT/CN2019/087000, filed on May 15, 2019, which is hereby incorporated by reference in its entirety.

TECHNICAL FIELD

This specification relates to determining action selection policies for an execution device for completing a task in an environment that includes the execution device and one or more other devices.

BACKGROUND

Strategic interaction between two or more parties can be modeled by a game that involves two or more parties (also referred to as players). In an Imperfect Information Game (IIG) that involves two or more players, a player only has partial access to the knowledge of her opponents before making a decision. This is similar to real-world scenarios, such as trading, traffic routing, and public auction. Many real life scenarios can be represented as IIGs, such as commercial competition between different companies, bidding relationships in auction scenarios, game relationships between a fraud party and an anti-fraud party.

Methods for solving an IIG are of great economic and societal benefits. Due to the hidden information, a player has to reason under the uncertainty regarding her opponents' information, and she also needs to act so as to take advantage of her opponents' uncertainty regarding her own information.

SUMMARY

This specification describes technologies for determining an action selection policy for an execution device for completing a task in an environment that includes the execution device and one or more other devices, for example, for strategic interaction between the execution device and the one or more other devices. For example, the execution device can perform a computer-implemented method for searching for a Nash equilibrium of a game between the execution device and one or more other devices. In some embodiments, these technologies can involve performing a fictitious counterfactual regret minimization (CFR) algorithm for solving an imperfect information game (IIG), which can save memory space, reduce the computational complexity and variance, while improving the convergence speed of the CFR algorithm.

This specification also describes one or more non-transitory computer-readable storage media, coupled to one or more processors and having instructions stored thereon which, when executed by the one or more processors, cause the one or more processors to perform operations in accordance with embodiments of the methods provided herein.

This specification further describes a system for implementing the methods described herein. The system includes one or more processors, and a computer-readable storage medium coupled to the one or more processors having instructions stored thereon which, when executed by the one or more processors, cause the one or more processors to perform operations in accordance with embodiments of the methods provided herein.

Methods, systems, and computer media in accordance with this specification may include any combination of the aspects and features described herein. That is, methods in accordance with this specification are not limited to the combinations of aspects and features specifically described herein, but also include any combination of the aspects and features described.

The details of one or more embodiments of this specification are set forth in the accompanying drawings and the description below. Other features and advantages of this specification will be apparent from the description and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating examples of partial game trees in one-card poker, in accordance with embodiments of this specification.

FIG. 2A is a diagram illustrating an example of a workflow of original CFR and streamline CFR, and FIG. 2B illustrates an example of a workflow of streamline CFR, in accordance with embodiments of this specification.

FIG. 3 is a pseudocode of an example of a streamline CFR algorithm, in accordance with embodiments of this specification.

FIG. 4 is a flowchart of an example of a process for performing a streamline CFR for determining action selection policies for software applications, in accordance with embodiments of this specification.

FIG. 5 is a diagram illustrating examples of an original CFR algorithm and a fictitious CFR algorithm applied on a partial game tree, in accordance with embodiments of this specification.

FIG. 6 is a flowchart of an example of a process for performing a fictitious CFR for strategy searching in strategic interaction between two or more parties, in accordance with embodiments of this specification.

FIG. 7 depicts a block diagram illustrating an example of a computer-implemented system used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures, in accordance with embodiments of this specification.

FIG. 8 is a diagram of an example of modules of an apparatus, in accordance with embodiments of this specification.

Like reference numbers and designations in the various drawings indicate like elements.

DETAILED DESCRIPTION

This specification describes technologies for determining an action selection policy for an execution device for completing a task in an environment that includes the execution device and one or more other devices, for example, for strategic interaction between the execution device and the one or more other devices. For example, the execution device can perform a computer-implemented method for searching for a Nash equilibrium of a game between the execution device and one or more other devices. In some embodiments, these technologies can involve performing a fictitious counterfactual regret minimization (CFR) algorithm for solving an imperfect information game (IIG), which can save memory space, reduce the computational complexity and variance, while improving the convergence speed of the CFR algorithm.

An IIG can represent one or more real-world scenarios such as resource allocation, product/service recommendation, cyber-attack prediction and/or prevention, traffic routing, fraud management, that involve two or more parties (also referred to as players), where each party may have incomplete or imperfect information about the other party's decisions.

Nash equilibrium is a typical solution for an IIG that involves two or more players. Counterfactual Regret Minimization (CFR) is an algorithm designed to approximately find Nash equilibrium for large games. CFR tries to minimize overall counterfactual regret. It is proven that the average of the strategies in all iterations would converge to a Nash equilibrium. When solving a game, CFR in its original form (also referred to as original CFR, standard CFR, vanilla CFR, or simply, CFR) traverses the entire game tree in each iteration. Thus, the original CFR requires large memory for large, zero-sum extensive games such as heads-up no-limit Texas Hold'em. In some instances, the original CFR may not handle large games with limited memory.

A Monte Carlo CFR (MCCFR) was introduced to minimize counterfactual regret. The MCCFR can compute an unbiased estimation of counterfactual value and avoid traversing the entire game tree. Since only subsets of all information sets are visited in each iteration, MCCFR requires less memory than the original CFR.

MCCFR can be performed with an outcome sampling algorithm or an external sampling algorithm. The outcome sampling algorithm in MCCFR has a large variance, and it is difficult to converge to an approximate Nash equilibrium solution in fewer iteration steps. The external sampling algorithm in MCCFR has a smaller variance than the outcome sampling algorithm, but this method presents similar disadvantages to CFR. When the game tree is large, it requires a very large memory space and cannot be extended to a complex large-scale IIG.

This specification discloses a streamline CFR algorithm. Compared to original CFR algorithm, in some embodiments, the space complexity of streamline CFR algorithm is about half of that of the original CFR algorithm. In some embodiments, the streamline CFR algorithm only needs one tabular memory or a single neutral network to track the key information while converging to comparable results produced by original CFR. The disclosed streamline CFR algorithm can be used in large games even with memory constraints. In some embodiments, the described techniques can be used, for example, in AI poker, recommendation platforms, and many other applications that can be modeled by a game that involves two or more parties.

CFR and its variants can use regret matching (RM) on trees to solve games. An RM algorithm can build strategies based on the concept of regret. For example, an RM algorithm can seek to minimize regret about its decisions at each step of a game. Compared to existing RM algorithms (e.g., original RM as described below with respect to Eq. (5)), this specification discloses a parameterized regret matching (PRM) algorithm with new parameters to reduce the variance of the original RM and decrease the computational load of the CFR algorithm.

Note that the PRM algorithm can be used not only in original CFR, but also its variants, including but not limited to, MCCFR and streamline CFR. In some embodiments, the PRM algorithm can be used not only in various CFR algorithms but also any other algorithms or techniques where the RM is applicable. For example, the PRM can be used to replace the original RM in algorithms other than the CFR algorithms to reduce the variance of the original RM and decrease the computational load of the other algorithms.

In some embodiments, an extensive-form game with a finite set N={0, 1, . . . , n−1} of players can be represented as follows: define h^(v) _(i) as a hidden variable of player i in an IIG. For example, in a poker game, h^(v) _(i) can refer to the private cards of player i. H refers to a finite set of histories. Each member h=(h_(i) ^(v))_(i=0, 1, . . . , n−1)(a_(l))_(l=0, . . . , L−1)=h₀ ^(v)h₁ ^(v) . . . h_(n−1) ^(v)a₀a₁ . . . a_(L−1) of H denotes a possible history (or state), which includes each player's hidden variable and L actions taken by players including chance. For player i, h also can be denoted as h_(i) ^(v)h_(−i) ^(v)a₀a₁ . . . a_(L−1), where h_(−i) ^(v) refers to the opponent's hidden variables. The empty sequence Ø is a member of H. The expression h_(j) ⊏h denotes that h_(j) is a prefix of h, where h_(j)=(h_(i) ^(v))_(i=0, 1, . . . , n−1)(a_(l))_(l=1, . . . L′−1) and 0<L′<L. Z⊆H denotes the terminal histories and any member z∈Z is not a prefix of any other sequences. A(h)={a:ha∈H} is the set of available actions after non-terminal history h∈H\Z. A player function P assigns a member of N∪{c} to each non-terminal history, where c denotes the chance player identifier (ID), which typically can be, for example, −1. P(h) is the player who takes an action after history h.

I_(i) of a history {h∈H:P(h)=i} is an information partition of player i. A set I_(i)∈I_(i) is an information set of player i. I_(i)(h) refers to information set I_(i) at state h. In some embodiments, I_(i) could only remember the information observed by player i including player i's hidden variable and public actions. Therefore I_(i) indicates a sequence in the IIG, i.e., h^(v) _(i) a₀a₂ . . . a_(L−1). In some embodiments, for I_(i)∈I_(i) and for any h∈I_(i), the set A(h) can be denoted by A(I_(i)) and the player P(h) is denoted by P(I_(i)). For each player i∈N, a utility function u_(i)(z) defines a payoff of a terminal state z. A more detailed explanation of these notations and definitions will be discussed below and will include an example shown in FIG. 1.

FIG. 1 is a diagram 100 illustrating examples of partial game trees 102 and 104 in One-Card Poker, in accordance with embodiments of this specification. One-Card Poker is a two-player IIG of poker. One-Card Poker is an example of an extensive-form game. The game rules are defined as follows. Each player is dealt one card from a deck of X cards. The first player can pass or bet. If the first player bets, the second player can call or fold. If the first player passes, the second player can pass or bet. If second player bets, the first player can fold or call. The game ends with two passes, a call, or a fold. The folding player will lose 1 chip. If the game ended with two passes, the player with the higher card wins 1 chip. If the game ends with a call, the player with the higher card wins 2 chips.

A game tree is a directed graph. The nodes of the game tree represent positions (or states of a player) in a game and of the game tree can represent moves or actions of a player of the game. In FIG. 1, z_(i) denotes a terminal node, representing a terminal state, and h_(i) denotes a non-terminal node. Each of the partial game trees 102 and 104 has a root node h₀ representing a chance. There are 19 distinct nodes in the first partial game tree 102, corresponding to 9 non-terminal nodes h_(i) including chance h₀ and 10 terminal nodes z_(i) in the left tree.

In the first partial game tree 102, two players (player 0 and player 1) are dealt (queen, jack) as shown as “0:Q 1:J” in the left subtree and (queen, king), as shown as “0:Q 1:K” in the right subtree.

The trajectory from the root node to each node is a history of actions. Actions are represented by letters (e.g., F, C, P, and B) or representations (e.g., “0:Q 1:J”) next to edges (denoted by arrows) of the game tree. The letters F, C, P, B refer to fold, call, pass, and bet, respectively.

In an extensive-form game, h_(i) refers to the history of actions. For example, as illustrated in the first partial game tree 102, h₃ includes actions 0:Q, 1:J and P. h₇ includes actions 0:Q, 1:J, P and B. h₈ includes actions 0:Q, 1:K, P and B. In the first partial game tree 102, h₃ ⊏h₇, that is, h₃ is a prefix of h₇. A(h₃)={P,B} indicating that the set of available actions after non-terminal history h₇ are P and B. P(h₃)=1 indicating that the player who takes an action after history h₃ is player 1.

In the IIG, the private card of player 1 is invisible to player 0, therefore h₇ and h₈ are actually the same for player 0. An information set can be used to denote the set of these undistinguished states. Similarly, h₁ and h₂ are in the same information set. For the right partial game tree 104, h′₃ and h′₅ are in the same information set; h′₄ and h′₆ are in the same information set.

Typically, any I_(i)∈I could only remember the information observed by player i including player i's hidden variables and public actions. For example, as illustrated in the first partial game tree 102, the information set of h₇ and h₈ indicates a sequence of 0:Q, P, and B. Because h₇ and h₈ are undistinguished by player 0 in the IIG, if I₀ is the information set of h₇ and h₈, I₀=I₀(h₇)=I₀(h₈).

A strategy profile σ={σ_(i)|σ_(i)∈Σ_(i), i∈N} is a collection of strategies for all players, where Σ_(i) is the set of all possible strategies for player i. σ_(−i) refers to strategy of all players other than player i. For player i∈N, the strategy σ_(i)(I_(i)) is a function, which assigns an action distribution over A(I_(i)) to information set I_(i). σ_(i)(a|h) which denotes the probability of action a taken by player i∈N∪{c} at state h. In an IIG, if two or more states have the same information set, the two or more states have a same strategy. That is, ∀h₁, h₂∈I_(i), I_(i)=I_(i)(h₁)=I_(i)(h₂), σ_(i)(I_(i))=σ_(i)(h₁)=σ_(i)(h₂), σ_(i)(a|I_(i))=σ_(i)(a|h₁)=σ_(i)(a|h₂). For example, I₀ is the information set of h₇ and h₈, I₀=I₀(h₇)=I₀(h₈), σ₀(I₀)=σ₀(h₇)=σ₀(h₈), σ₀(a|I₀)=σ₀(a|h₇)=σ₀(a|h₈). In FIG. 1, the same shading (other than the gray ones) is used to represent the same information set in respective state.

For player i, the expected game utility of the strategy profile σ is denoted as u_(i) ^(σ)=Σ_(z∈Z)π^(σ)(z)u_(i)(z), which is the expected payoff of all possible terminal nodes. Given a fixed strategy profile σ_(−i), any strategy σ_(i)*=arg max_(σ′) _(i) _(∈Σ) _(i) u_(i) ^((σ′) ^(i) ^(,σ−i)) of player i that achieves maximize payoff against π_(−i) ^(σ) is a best response. For two players' extensive-form games, a Nash equilibrium is a strategy profile σ*=(σ*₀,σ*₁) such that each player's strategy is a best response to the opponent. An ϵ-Nash equilibrium is an approximation of a Nash equilibrium, whose strategy profile σ* satisfies: ∀_(i)∈N, u_(i) ^(σ) ^(i) +ϵ≥max_(σ′) _(i) _(∈Σ) _(i) u_(i) ^((σ′) ^(i) ^(,σ−i)).

Exploitability of a strategy σ_(i) can be defined as ϵ_(i)(σ_(i))=u_(i) ^(σ*)−u_(i) ^((σ) ^(i) ^(,σ*) ^(−i) ⁾. A strategy is unexploitable if ϵ_(i)(σ_(i))=0. In large two player zero-sum games such as poker, u_(i) ^(σ*) can be intractable to compute. However, if the players alternate their positions, the value of a pair of games is zero, i.e., u₀ ^(σ*)+u₁ ^(σ*)=0. The exploitability of strategy profile σ can be defined as ϵ(σ)=(u₁ ^((σ) ⁰ ^(,σ*) ¹ ⁾+u₀ ^((σ*) ⁰ ^(,σ) ¹ ⁾/2.

For iterative methods such as CFR, σ^(t) can refer to the strategy profile at the t-th iteration. The state reach probability of history h can be denoted by π^(σ)(h) if players take actions according to σ. For an empty sequence π^(σ)(Ø)=1. The reach probability can be decomposed into π^(σ)(h)=Π_(i∈N∪{c})π_(i) ^(σ)(h)=π_(i) ^(σ)(h)π_(−i) ^(σ)(h) according to each player's contribution, where π_(i) ^(σ)(h)=Π_(h′a⊏h,P(h′)=P(h′))σ_(i)(a|h′) and π_(−i) ^(σ)(h)=Π_(h′a⊏h,P(h′)≠P(h))σ_(−i)(a|h′).

The reach probability of information set I_(i) (also referred to as information set reach probability) can be defined as π^(σ)(I_(i))=Σ_(h∈I) _(i) π^(σ)(h). If h′⊏h, the interval state reach probability from state h′ to h can be defined as π^(σ)(h′,h), then π^(σ)(h′,h)=π^(σ)(h)/π^(σ)(h′). The reach probabilities π_(i) ^(σ)(I_(i)), π_(i) ^(σ)(h′, h), and π_(−i) ^(σ)(h′, h) can be defined similarly.

In large and zero-sum IIGs, CFR is proved to be an efficient method to compute Nash equilibrium. It is proved that the state reach probability of one player is proportional to posterior probability of the opponent's hidden variable, i.e., p(h_(−i) ^(v)|I_(i))∝π_(−i) ^(σ)(h), where h^(v) _(i) and I_(i) indicate a particular h.

For player i and strategy profile σ, the counterfactual value (CFV) v_(i) ^(σ)(h) at state h can be defined as: v _(i) ^(σ)(h)=Σ_(h⊏z,z∈Z)π_(−i) ^(σ)(h)π^(σ)(h,z)u _(i)(z)=Σ_(h⊏z,z∈Z)π_(i) ^(σ)(h,z)u′ _(i)(z)  (1)

where u′_(i)(z)=π_(−i) ^(σ)(z)u_(i)(z) is the expected reward of player i with respect to the approximated posterior distribution of the opponent's hidden variable. Then the counterfactual value of information set I_(i) is v_(i) ^(σ)(I_(i))=Σ_(h∈I) _(i) v_(i) ^(σ)(h).

The action counterfactual value of taking action a can be denoted as v_(i) ^(σ)(a|h)=v_(i) ^(σ)(ha) and the regret of taking this action is: r _(i) ^(σ)(a|h)=v _(i) ^(σ)(a|h)−v _(i) ^(σ)(h)  (2).

Similarly, the CFV of information set I_(i) can be defined as v_(i) ^(σ)(I_(i))=Σ_(h∈I) _(i) v_(i) ^(σ)(h), while the CFV of its action a is v_(i) ^(σ)(a|I_(i))=Σ_(z∈Z,h⊏z,h∈I) _(i) π_(i) ^(σ)(h, z)u′_(i)(z) and the regret of action a given the information set I_(i) can be defined as: r _(i) ^(σ)(a|I _(i))=v _(i) ^(σ)(a|I _(i))−v _(i) ^(σ)(I _(i))=Σ_(z∈Z,ha⊏z,h∈I) _(i) π_(i) ^(σ)(ha,z)u′ _(i)(z)−Σ_(z∈Z,h⊏z,h∈I) _(i) π_(i) ^(σ)(h,z)u′ _(i)(z),  (3)

where

${u_{i}^{\sigma}\left( I_{i} \right)} = {\frac{\sum\limits_{h \in I_{i}}{v_{i}^{\sigma}(h)}}{\sum\limits_{h \in I_{i}}{\pi_{- i}^{\sigma}(h)}} = {\frac{\sum\limits_{h \in I_{i}}{v_{i}^{\sigma}(h)}}{\pi_{- i}^{\sigma}\left( I_{i} \right)}.}}$ Note that, in imperfect information game, π_(−i) ^(σ)(I_(i))=π_(−i) ^(σ)(h).

Then, the accumulative regret of action a after T iterations can be calculated or computed according to Eq. (4):

$\begin{matrix} {{R_{i}^{T}\left( a \middle| I_{i} \right)} = {{\sum\limits_{t = 1}^{T}\left( {{v_{i}^{\sigma^{t}}\left( a \middle| I_{i} \right)} - {v_{i}^{\sigma^{t}}\left( I_{i} \right)}} \right)} = {{R_{i}^{T - 1}\left( a \middle| I_{i} \right)} + {r_{i}^{\sigma^{T}}\left( a \middle| I_{i} \right)}}}} & (4) \end{matrix}$ where R_(i) ⁰(a|I_(i))=0. Defining R_(i) ^(T,+)(a|I_(i))=max(R_(i) ^(T)(a|I_(i)),0), the current strategy (or iterative strategy or behavior strategy) at T+1 iteration can be updated, for example, based on regret matching (RM), according to Eq. (5) below:

$\begin{matrix} {{\sigma_{i}^{T + 1}\left( a \middle| I_{i} \right)} = \left\{ {\begin{matrix} {\frac{R_{i}^{T, +}\left( a \middle| I_{i} \right)}{\sum\limits_{a \in {A{(I_{i})}}}{R_{i}^{T, +}\left( a \middle| I_{i} \right)}},} & {{{if}\mspace{14mu}{\sum\limits_{a \in {A{(I_{i})}}}{R_{i}^{T, +}\left( a \middle| I_{i} \right)}}} > 0} \\ {\frac{1}{{A\left( I_{i} \right)}},} & {otherwise} \end{matrix}.} \right.} & (5) \end{matrix}$

The average strategy σ _(i) ^(T) from iteration 1 to T can be defined as:

$\begin{matrix} {{{\overset{¯}{\sigma}}_{i}^{T}\left( a \middle| I_{i} \right)} = \frac{\sum\limits_{t = 1}^{T}{{\pi_{i}^{\sigma^{t}}\left( I_{i} \right)}{\sigma_{i}^{t}\left( a \middle| I_{i} \right)}}}{\sum\limits_{t = 1}^{T}{\pi_{i}^{\sigma^{t}}\left( I_{i} \right)}}} & (6) \end{matrix}$ where π_(i) ^(σ) ^(t) (I_(i)) denotes the information set reach probability of I_(i) at t-th iteration and is used to weigh the corresponding current strategy σ_(i) ^(t)(a|I_(i)).

If s^(t)(a|I_(i))=π_(i) ^(σ) ^(t) (I_(i))σ_(i) ^(t)(a|I_(i)) is defined as an additional numerator in iteration t, then the accumulative numerator of the average strategy σ _(i) ^(T) can be defined as:

$\begin{matrix} {{{S^{T}\left( a \middle| I_{i} \right)} = {{\sum\limits_{t = 1}^{T}{{\pi_{i}^{\sigma^{t}}\left( I_{i} \right)}{\sigma_{i}^{t}\left( a \middle| I_{i} \right)}}} = {{S^{T - 1}\left( a \middle| I_{i} \right)} + {s_{i}^{T}\left( a \middle| I_{i} \right)}}}},} & (7) \end{matrix}$ where S⁰(a|I_(i))=0.

For the streamline CFR, unlike the iterative strategy σ_(i) ^(T+)1 (a|I_(i)) in the original CFR, an incremental strategy σ̆_(i) ^(t+1)(a|I_(i)) is defined as in Eq. (8):

$\begin{matrix} {{{\overset{\Cup}{\sigma}}_{i}^{t + 1}\left( a \middle| I_{i} \right)} = \left\{ {\begin{matrix} {\frac{{\overset{\Cup}{R}}_{i}^{t, +}\left( a \middle| I_{i} \right)}{\sum\limits_{a \in {A{(I_{i})}}}{{\overset{\Cup}{R}}_{i}^{t, +}\left( a \middle| I_{i} \right)}},} & {{{if}\mspace{14mu}{\sum\limits_{a \in {A{(I_{i})}}}{{\overset{\Cup}{R}}_{i}^{t, +}\left( a \middle| I_{i} \right)}}} > 0} \\ {\frac{1}{{A\left( I_{i} \right)}},} & {otherwise} \end{matrix},} \right.} & (8) \end{matrix}$ wherein R̆^(t)(a|I_(i))=r_(i) ^(σ) ^(t) (a|I_(i)), and σ̆¹=(σ̆_(i) ¹,σ̆_(−i) ¹) is an initial strategy, for example, initialized by a random policy, such as a uniform random strategy profile, or another initialization policy.

The iterative strategy of the streamline CFR in iteration t can be defined by Eq. (9): σ_(i) ^(t)(a|I _(i))=(1−α^(t)(I _(i)))σ_(i) ^(t−1)(a|I _(i))+α^(t)(I _(i))σ̆_(i) ^(t)(a|I _(i))  (9) where α^(t)(I_(i)) is the learning rate for I_(i) in t-th iteration and σ_(i) ⁰(a|I_(i))=0. The learning rate α^(t)(I_(i)) approaches 0 as t approaches infinity. As an example, α^(t)(I_(i)) can be set as 1/t or another value. With Eq. (9), the iterative strategy in the next iterations can be obtained. After enough iterations, the iterative strategy profile σ_(i) ^(T)(a|I_(i)) obtained by the streamline CFR can converge to an approximated Nash equilibrium. It is proved that the iterative strategy profile defined by Eq. (9) can converge to a set of Nash equilibria in two-player zero-sum games.

In some embodiments, in a fictitious CFR, unlike the CFV in the original CFR as defined in Eq. (1), a fictitious counterfactual value (FCFV) can be defined as: v _(i) ^(σ̆) ^(t+1) (I _(i))=Σ_(a∈A(I) _(i) ₎σ̆_(i) ^(t+1)(a|I _(i))v _(i) ^(t)(a|I _(i))  (10)

In some embodiments, the CFV in the original CFR as defined in Eq. (1) can be rewritten as v _(i) ^(σ) ^(t) (I _(i))=Σ_(a∈A(I) _(i) ₎σ_(i) ^(t)(a|I _(i))v _(i) ^(t)(a|I _(i))  (11) which is a weighted summation from leaf nodes to a root node of a game tree recursively. For the FCFV in Eq. (10), v_(i) ^(σ̆) ^(t+1) (I_(i)) is used to replace v_(i) ^(σ) ^(t) (I_(i)), i.e., v_(i) ^(σ) ^(t) (I_(i))=v_(i) ^(σ̆) ^(t+1) (I_(i)). After the incremental strategy of I_(i), σ̆_(i) ^(t+1)(a|I_(i)), has been calculated, for example, according to Eq. (8). For example, v_(i) ^(σ)(a|I_(i)) can be used to calculate r_(i) ^(σ)(a|I_(i)). Then Eq. (8) can be used to update incremental strategy σ̆_(i) ^(t+1)(a|I_(i)). In some embodiments, only the counterfactual value of those nodes whose children nodes have all been visited can be replaced by the FCFV. FIG. 5 shows an example updating process of a FCFV, compared to the CFV of a node in a game tree.

In some embodiments, the FCFV v_(i) ^(σ̆) ^(t+1) (I_(i)) as defined in Eq. (10) can be calculated right after the incremental strategy of I_(i), σ̆_(i) ^(t+1)(a|I_(i)) has been calculated, for example, according to Eq. (8). After that, v_(i) ^(σ) ^(t) (I_(i))=v_(i) ^(σ̆) ^(t+1) (I_(i)), that is, the FCFV is used as the CFV. Therefore, there is no need for memory to store v_(i) ^(σ̆) ^(t+1) (I_(i)). As such, the memory efficiency and space complexity of the fictitious CFR algorithm can be improved compared to the original CFR algorithm.

When solving a game, the original CFR traverses the entire game tree in each iteration. Thus, the original CFR may not handle large games with limited memory. A Monte Carlo CFR (MCCFR) was introduced to minimize counterfactual regret. The MCCFR can compute an unbiased estimation of counterfactual value and avoid traversing the entire game tree. Since only subsets of all information sets are visited in each iteration, MCCFR requires less memory than the original CFR.

For example, define Q={Q₁, Q₂, . . . , Q_(m)}, where Q_(j)∈Z is a block of sampling terminal histories in each iteration, such that Q_(j) spans the set Z. Generally, different Q_(j) may have an overlap according to a specified sampling scheme. Several sampling schemes can be used.

FIG. 2A is a diagram illustrating an example of a workflow 200 of original CFR and streamline CFR, and FIG. 2B illustrates an example of a workflow 205 of streamline CFR, in accordance with embodiments of this specification. As illustrated, both the original CFR and the streamline CFR can be performed in an iterative manner. FIGS. 2A and B show four iterations, t=1, 2, 3 or 4, respectively. The superscript 1, 2, 3, or 4 represents the t-th iteration. The original CFR and the streamline CFR can include more iterations. To simplify the expression, the subscript i is omitted under each of: R_(i) ^(t)(a|I_(i)), σ_(i) ^(t)(a|I_(i)), σ̆_(i) ^(t)(a|I_(i)), and r_(i) ^(σ) ^(i) ^(t) (a|I_(i)).

As illustrated in the workflow 205 of the streamline CFR in FIG. 2B, in the first iteration, t=1, an incremental strategy σ̆¹(a|I) 213 can be computed based on an initial regret value r^(σ) ⁰ (a|I) 211, for example, according to Eq. (8). The iterative strategy σ¹(a|I) 215 can be computed based on the incremental strategy σ̆¹(a|I) 213 and an initial iterative strategy σ⁰(a|I)=0, for example, according to Eq. (9). Based on the iterative strategy σ¹(a|I) 215, an updated regret value of the iterative strategy of r^(σ) ¹ (a|I) 221 can be computed, for example, according to Eq. (3) based on the counterfactual values by traversing the game tree recursively.

The updated regret value of the iterative strategy of r^(σ) ¹ (a|I) 221 can be used to compute an updated incremental strategy σ̆²(a|I) 223 in the next iteration, t=2, for example, according to Eq. (8). The iterative strategy σ²(a|I) 225 can be computed based on the incremental strategy σ̆²(a|I) 223 and the iterative strategy σ¹ (a|I) 215 in the first iteration, for example, according to Eq. (9). Similarly, based on the iterative strategy σ²(a|I) 225, an updated regret value r^(σ) ² (a|I) 231 of the iterative strategy σ²(a|I) 225 can be computed, for example, according to Eq. (3) based on the counterfactual values by traversing the game tree recursively.

Similarly, in the next iteration, t=3, based on the updated regret value r^(σ) ² (a|I) 231, an updated incremental strategy σ̆³(a|I) 233 can be computed, for example, according to Eq. (8). An iterative strategy σ³(a|I) 235 can be computed based on the incremental strategy σ̆³(a|I) 233 and the iterative strategy σ²(a|I) 225, for example, according to Eq. (9). Based on the iterative strategy σ³(a|I) 235, an updated regret value r^(σ) ³ (a|I) 241 of the iterative strategy σ³(a|I) 235 can be computed, for example, according to Eq. (3) based on the counterfactual values by traversing the game tree recursively.

In the next iteration, t=4, based on the updated regret value r^(σ) ³ (a|I) 241, an updated incremental strategy σ̆⁴(a|I) 243 can be computed, for example, according to Eq. (8). An iterative strategy σ⁴(a|I) 245 can be computed based on the incremental strategy σ̆⁴(a|I) 244 and the iterative strategy σ³(a|I) 235, for example, according to Eq. (9). Based on the iterative strategy σ⁴(a|I) 245, an updated regret value r^(σ) ⁵ (a|I) (not shown) of the iterative strategy σ⁴(a|I) 245 can be computed, for example, according to Eq. (4) based on the counterfactual values by traversing the game tree recursively. The updated regret value r^(σ) ⁵ (a|I) can be used for computing an incremental strategy for the next iteration. The streamline CFR can repeat the above iterations until convergence is achieved.

Note that in the streamline CFR, as illustrated in FIG. 2B, an incremental strategy in a current iteration (e.g., σ̆^(T)(a|I) in the T-th iteration) can be computed based on a regret value of the action in an immediately previous iteration (e.g., r^(σ) ^(T−1) (a|I) in the (T−1)th iteration) but not any regret value of the action in any other previous iteration (e.g., (T−2)th iteration, (T−3)th iteration). And the iterative strategy in a current iteration (e.g., σ^(T)(a|I) in the T-th iteration) can be computed based on the iterative strategy of the action in the (T−1)-th iteration (e.g., σ^(T−1)(a|I) in the (T−1)-th iteration) and the incremental strategy of the action in the current iteration (e.g., σ̆^(T)(a|I) in the t-th iteration). As such, only the iterative strategy in the current iteration (e.g., σ̆^(T)(a|I) in the T-th iteration) needs to be stored for computing an updated iterative strategy in the next iteration (e.g., σ^(T+1)(a|I) in the (T+1)-th iteration). This is in contrast to the original CFR. For example, for a current iteration (e.g., T-th iteration), the original CFR proceeds based on a cumulative regret R_(i) ^(T)(a|I_(i)) and average strategy σ _(i) ^(T) over all t=1, 2, . . . , T iterations,

As illustrated in the workflow 200 of the original CFR in FIG. 2A, in the first iteration, t=1, an iterative strategy σ¹(a|I) 214 can be computed based on an initial accumulative regret R⁰(a|I) 212, for example, according to Eq. (5). An average strategy σ ¹(a|I) 210 can be computed based on the iterative strategy σ¹(a|I) 214 and an initial average strategy σ⁰(a|I)=0, for example, according to Eq. (6). Based on the iterative strategy σ¹(a|I) 214, an updated regret value of the iterative strategy of r^(σ) ¹ (a|I) 216 can be computed, for example, according to Eq. (3) based on the counterfactual values by traversing the game tree recursively. An updated accumulative regret R¹(a|I) 222 of action a after the first iteration can be computed based on the iterative strategy of r^(σ) ¹ (a|I) 216 and the initial accumulative regret R⁰(a|I) 212, for example, according to Eq. (4).

In the second iteration, t=2, an iterative strategy σ²(a|I) 224 can be computed based on the updated accumulative regret R¹(a|I) 222, for example, according to Eq. (5). An average strategy σ ²(a|I) 220 can be computed based on the iterative strategy σ²(a|I) 224 and the average strategy σ ¹(a|I) 210 in the first iteration, for example, according to Eq. (6). Based on iterative strategy σ²(a|I) 224, an updated regret value of the iterative strategy of r^(σ) ² (a|I) 226 can be computed, for example, according to Eq. (3) based on the counterfactual values, by traversing the game tree recursively. An updated accumulative regret R²(a|I) 232 of action a after the second iteration can be computed based on the iterative strategy of r^(σ) ² (a|I) 226 and the accumulative regret R¹(a|I) 222, for example, according to Eq. (4).

In the third iteration, t=3, an iterative strategy σ³(a|I) 234 can be computed based on the updated accumulative regret R²(a|I) 232, for example, according to Eq. (5). An average strategy σ ³(a|I) 230 can be computed based on the iterative strategy σ³(a|I) 234 and the average strategy σ ²(a|I) 220 in the second iteration, for example, according to Eq. (6). Based on iterative strategy σ³(a|I) 234, an updated regret value of the iterative strategy of r^(σ) ³ (a|I) 236 can be computed, for example, according to Eq. (3) based on the counterfactual values, by traversing the game tree recursively. An updated accumulative regret R³(a|I) 242 of action a after the third iteration can be computed based on the iterative strategy of r^(σ) ³ (a|I) 236 and the accumulative regret R²(a|I) 232, for example, according to Eq. (4).

In the fourth iteration, t=4, an iterative strategy σ⁴(a|I) 244 can be computed based on the updated accumulative regret R³(a|I) 242, for example, according to Eq. (5). An average strategy σ ⁴(a|I) 240 can be computed based on the iterative strategy σ⁴(a|I) 244 and the average strategy σ ³(a|I) 230 in the third iteration, for example, according to Eq. (6). Based on iterative strategy σ⁴(a|I) 244, an updated regret value of the iterative strategy of r^(σ) ⁴ (a|I) (not shown) can be computed, for example, according to Eq. (3) based on the counterfactual values, by traversing the game tree recursively. Similarly, an updated accumulative regret R⁴(a|I) (not shown) of action a after the fourth iteration can be computed based on the iterative strategy of r^(σ) ⁴ (a|I) and the accumulative regret R³(a|I) 242, for example, according to Eq. (4). The original CFR can repeat the above iterations until convergence is achieved.

As illustrated in the workflow 200 of the original CFR in FIG. 2A, the original CFR needs to track at least two values in each iteration, that is, the accumulative regret R_(i) ^(T)(a|I_(i)) and the average strategy σ _(i) ^(T) over all t=1, 2, . . . , T iterations, as each iteration of the original CFR relies not only on the regret and strategy of the immediately preceding iteration but also on those in all iterations prior to the immediately preceding iteration. On the other hand, each iteration of the streamline CFR can proceed without the knowledge of any regret values or strategies in any iteration prior to the immediately preceding iteration (e.g., (T−2)th iteration, (T−3)th iteration). For example, the streamline CFR may only need to store the iterative strategies (e.g., σ¹(a|I) 215, σ²(a|I) 225, σ³(a|I) 235, σ⁴(a|I) 245) as shown as gray blocks in FIG. 2A), whereas the original CFR needs to store accumulative regrets (e.g., R⁰(a|I) 212, R¹(a|I) 222, R²(a|I) 232 and R³(a|I) 242) as well as average strategies (e.g., σ ¹(a|I) 210, σ ²(a|I) 220, σ ³(a|I) 230, and σ ⁴(a|I) 240 shown as gray blocks in FIG. 2B) in each iteration. As such, the streamline CFR requires less storage space than the original CFR (e.g., half of the storage space), providing improved memory efficiency.

FIG. 3 is a pseudocode 300 of an example of a streamline CFR algorithm, in accordance with embodiments of this specification. In some embodiments, a streamline CFR algorithm is an iterative algorithm. Within each iteration t, a function SCFR is called for player 0 and player 1 to update an incremental strategy σ̆_(i)(I_(i)) and an iterative strategy σ_(i) ^(t+1)(I_(i)) as shown in lines 25 and 26 of the pseudocode 300, respectively. The incremental strategy σ̆_(i)(I_(i)) is updated using a function CalculateStrategy as defined in lines 29-33 of the pseudocode 300. The function CalculateStrategy is an example implementation of Eq. (8). The iterative strategy σ_(i) ^(t+1)(I_(i)) can be updated according to Eq. (9). The function SCFR returns the counterfactual value of each information set as the output, which is computed by traversing the game tree recursively as shown in lines 4-27 of the pseudocode 300.

FIG. 4 is a flowchart of an example of a process for performing a streamline counterfactual regret minimization (CFR) for determining action selection policies for software applications, for example, for strategy searching in strategic interaction between two or more parties, in accordance with embodiments of this specification. The process 400 can be an example of the streamline CFR algorithm described above with respect to FIGS. 2-3. In some embodiments, the process 400 can be performed in an iterative manner, for example, by performing two or more iterations. In some embodiments, action selection policies can be determined for software applications, for example, in strategic interaction between two or more players can be modeled by an imperfect information game (IIG) that involves two or more players. In some embodiments, the process 400 can be performed for solving an IIG. The IIG can represent one or more real-world scenarios such as resource allocation, product/service recommendation, cyber-attack prediction and/or prevention, traffic routing, fraud management, etc. that involves two or more parties, where each party may have incomplete or imperfect information about the other party's decisions. As an example, the IIG can represent a collaborative product-service recommendation service that involves at least a first player and a second player. The first player may be, for example, an online retailer that has customer (or user) information, product and service information, purchase history of the customers, etc. The second player can be, for example, a social network platform that has social networking data of the customers, a bank or another finical institution that has financial information of the customers, a car dealership, or any other parties that may have information of the customers on the customers' preferences, needs, financial situations, locations, etc. in predicting and recommendations of products and services to the customers. The first player and the second player may each have proprietary data that the player does not want to share with others. The second player may only provide partial information to the first player at different times. As such, the first player may only have limited access to information of the second player. In some embodiments, the process 400 can be performed for making a recommendation to a party with limited information of the second party, such as planning a route with limited information.

For convenience, the process 400 will be described as being performed by a data processing apparatus such as a system of one or more computers, located in one or more locations, and programmed appropriately in accordance with this specification. For example, a computer system 700 of FIG. 7, appropriately programmed, can perform the process 400.

At 402, an iterative strategy of an action in a state of a party in a first iteration, i.e., t=1 iteration, is initialized. In some embodiments, the iterative strategy can be initialized, for example, based on an existing strategy, a uniform random strategy (e.g. a strategy based on a uniform probability distribution), or another strategy (e.g. a strategy based on a different probability distribution). For example, if the system warm starts from an existing CFR method (e.g., an original CFR or MCCFR method), the iterative strategy can be initialized from an existing strategy profile to clone existing regrets and strategy.

In some embodiments, the strategic interaction between two or more parties can be modeled by an imperfect information game (IIG). As an example, the IIG represents a collaborative product-service recommendation service that involves the party and a second party. The party has limited access to information of the second party. The state of the party comprises a history of information provided by the second party, and the action of the party comprises an action in response to the history of information provided by the second party for providing product-service recommendations to customers.

At 404, whether a convergence condition is met is determined. The convergence condition can be used for determining whether to continue or terminate the iteration. In some embodiments, the convergence condition can be based on exploitability of a strategy σ. According to the definition of exploitability, exploitability should be large than or equal to 0. The smaller exploitability indicates a better strategy. That is, the exploitability of converged strategy should approach 0 after enough iterations. For example, in poker, when the exploitability is less than 1, the time-average strategy is regarded as a good strategy and it is determined that the convergence condition is met. In some embodiments, the convergence condition can be based on a predetermined number of iterations. For example, in a small game, the iterations can be easily determined by the exploitability. That is, if exploitability is small enough, the process 400 can terminate. In a large game, the exploitability is intractable, typically a large parameter for iteration can be specified. After each iteration, a new strategy profile can be obtained, which is better than the old one. For example, in a large game, the process 400 can terminate after a sufficient number of iterations.

If the convergence condition is met, no further iteration is needed. The process 400 proceeds to 416, where an iterative strategy (the latest strategy in the current iteration) is outputted. If the convergence condition is not met, t is increased by 1, and the process 400 proceeds to a next iteration, wherein t>1.

In a current iteration (e.g., t-th iteration), at 406, an iterative strategy of an action in a state of a party in a (t−1)-th iteration (e.g., an iterative strategy σ_(i) ^(t−1)(a|I_(i)) of an action a in a state of a party represented by an information set I_(i) in a (t−1)-th iteration) is identified. The iterative strategy of the action in the state of the party in the (t−1)-th iteration represents a probability of the action taken by the party in the state in the (t−1)-th iteration.

At 408, a regret value of the action in the state of the party in the (t−1)-th iteration (e.g., r_(i) ^(σ) ^(t−1) (a|I_(i))) is computed based on the iterative strategy of the action in the state of the party in the (t−1)-th iteration. In some embodiments, computing a regret value of the action in the state of the party in the (t−1)-th iteration based on the iterative strategy of the action in the state of the party in the (t−1)-th iteration comprises computing the regret value of the action in the state of the party in the (t−1)-th iteration based on the iterative strategy of the action in the state of the party in the (t−1)-th iteration but not any regret value of the action in the state of the party in any iteration prior to the (t−1)-th iteration.

In some embodiments, computing a regret value of the action in the state of the party in the (t−1)-th iteration based on the iterative strategy of the action in the state of the party in the (t−1)-th iteration comprises computing the regret value of the action in the state of the party in the (t−1)-th iteration based on a difference between a counterfactual value of the action in the state of the party and a counterfactual value of the state of the party (e.g., according to Eq. (3)), wherein the counterfactual value of the action in the state of the party and the counterfactual value of the state of the party are computed by recursively traversing a game tree that represents the strategic interaction between the two or more parties in the (t−1)-th iteration (e.g., as shown in lines 4-27 of the pseudocode 300 in FIG. 3).

At 410, an incremental strategy of the action in the state of the party in the t-th iteration (e.g., σ̆_(i) ^(t+1)(a|I_(i))) is computed based on the regret value of the action in the state of the party in the (t−1)-th iteration but not any regret value of the action in the state of the party in any iteration prior to the (t−1)-th iteration. In some embodiments, the incremental strategy of the action in the state of the party in the t-th iteration is computed based on the regret value of the action in the state of the party in the (t−1)-th iteration but not any regret value of the action in the state of the party in any iteration prior to the (t−1)-th iteration according to Eq. (8). For example, the incremental strategy of the action in the state of the party in the t-th iteration is computed based on the regret value of the action in the state of the party in the (t−1)-th iteration but not any regret value of the action in the state of the party in any iteration prior to the (t−1)-th iteration according to:

${{\overset{\Cup}{\sigma}}_{i}^{t}\left( a \middle| I_{i} \right)} = \left\{ {\begin{matrix} {\frac{{\overset{\Cup}{R}}_{i}^{{t - 1}, +}\left( a \middle| I_{i} \right)}{\sum\limits_{a \in {A{(I_{i})}}}{{\overset{\Cup}{R}}_{i}^{{t - 1}, +}\left( a \middle| I_{i} \right)}},} & {{{if}\mspace{14mu}{\sum\limits_{a \in {A{(I_{i})}}}{{\overset{\Cup}{R}}_{i}^{{t - 1}, +}\left( a \middle| I_{i} \right)}}} > 0} \\ {\frac{1}{{A\left( I_{i} \right)}},} & {otherwise} \end{matrix},} \right.$

wherein a represents the action, I_(i) represents the state of the party, σ̆_(i) ^(t)(a|I_(i)) represents the incremental strategy of the action in the state of the party in the t-th iteration, R̆^(t−1)(a|I_(i))=r_(i) ^(σ) ^(t) (a|I_(i)) represents the regret value of the action in the state of the party in the (t−1)-th iteration, R̆_(i) ^(t−1,+)(a|I_(i))=max(R̆_(i) ^(t−1)(a|I_(i)),0), and |A(I_(i))| represents a number of total available actions in the state of the party.

At 412, an iterative strategy of the action in the state of the party in the t-th iteration is computed based on a weighted sum of the iterative strategy of the action in the state of the party in the (t−1)-th iteration and the incremental strategy of the action in the state of the party in the t-th iteration. For example, the iterative strategy of the action in the state of the party in the t-th iteration is computed based on a weighted sum of the iterative strategy of the action in the state of the party in the (t−1)-th iteration and the incremental strategy of the action in the state of the party in the t-th iteration according to Eq. (9). In some embodiments, the weighted sum of the iterative strategy of the action in the state of the party in the (t−1)-th iteration and the incremental strategy of the action in the state of the party in the t-th iteration comprises a sum of the iterative strategy of the action in the state of the party in the (t−1)-th iteration scaled by a first learning rate in the t-th iteration and the incremental strategy of the action in the state of the party in the t-th iteration scaled by a second learning rate in the t-th iteration. The first learning rate approaches 1 as t approaches infinity, and the second learning rate approaches 0 as t approaches infinity. In some embodiments, the first learning rate is (t−1)/t, and the second learning rate is 1/t.

At 414, the iterative strategy of the action in the state of the party in the t-th iteration is stored, for example, for computing the iterative strategy of the action in the state of the party in the (t+1)-th iteration. In some embodiments, the iterative strategy of the action in the state of the party in the t-th iteration can be stored in a memory (e.g., in a table or another data structure in a memory) or another data store. In some embodiments, the iterative strategy of the action in the state of the party in the t-th iteration can be stored by a neutral network. For example, a neutral network can be used to learn the iterative strategy of the action in the state of the party in the t-th iteration, for example, for predicting the iterative strategy of the action in the state of the party in the (t+1)-th iteration. In some embodiments, compared to the original CFR, the streamline CFR algorithm only needs a half of the storage size or a single rather than double neutral network to track the key information while converging to comparable results produced by the original CFR.

At 416, in response to determining that a convergence condition is met, the iterative strategy of the action in the state of the party in the t-th iteration is outputted. In some embodiments, the iterative strategy of the action in the state of the party in the t-th iteration can be used to approximate Nash equilibrium and serve as an output of the CFR algorithm. In some embodiments, the iterative strategy of the action in the state of the party can include a series of actions of the player in the real-world scenario modeled by the IIG. For example, in the collaborative product-service recommendation scenario, the iterative strategy of the action in the state of the party can include, for example, a series of actions in response to the information provided by the second player, corresponding product-service recommendations to customers based on the information of the first player and the information provided by the second player. The iterative strategy of the action in the state of the party can include other information in other real-world scenarios that are modeled by the IIG.

FIG. 5 is a diagram illustrating examples 500 a and 500 b of an original CFR algorithm and a fictitious CFR algorithm applied on a partial game tree, respectively, in accordance with embodiments of this specification. In both examples 500 a and 500 b, the partial game tree includes nodes 0, 1, 2, . . . , and 7, where the node 0 is a root node and nodes 6 and 7 are the leaf nodes. The node i correspond to an information set I^(i).

The example 500 a shows the CFV updating process of an original CFR algorithm. In each iteration, the original CFR needs to maintain a current strategy, and use the current strategy to generate the CFV (e.g., according to Eq. (1) or (11)), and use the regret matching algorithm (e.g., according to Eq. (5)) to calculate a current strategy of the next iteration. The weighted average of the current strategies for all iterative processes can converge to a Nash Equilibrium.

The example 500 b shows a FCFV updating process of a fictitious CFR algorithm. In some embodiments, in each iteration, the fictitious CFR algorithm can traverse the entire game tree, and uses a bottom-up process to update the CFV of each node of the tree. For example, as shown in 500 b, since node 6 and node 7 are leaf nodes, their CFVs can be considered as their respective FCFVs. For node 5, the CFV of node 5, v(I⁵), can be calculated based on strategy σ^(t), for example, according to v(I⁵)=Σ_(a)v(a|I⁵)σ(a|I⁵)=v(a₆|I⁵)σ^(t)(a₆|I⁵)+v(a₇|I⁵)σ^(t)(a₇|I⁵), wherein v(a₆|I⁵) is the CFV of the action a₆ leading to node 6 and v(a₇|I⁵) is the CFV of the action a₇ leading to the node 7.

The regret values of the node 5 can be calculated based on the CFV of node 5 v(I⁵). For example, the regret value of action a₆ in the state of the node 5 can be calculated according to r(a₆|I⁵)=v(a₆|I⁵)−v(I⁵). Similarly, the regret value of action a₇ in the state of the node 5 can be calculated according to r(a₇|I⁵)=v(a₆|I⁵)−v(I⁵).

Based on the regret values of the node 5, incremental strategies of the two actions a₆ and a₇ at node 5 can be calculated, for example, according to the regret matching of the streamline CFR algorithm as shown in Eq. (8). In some embodiments, the incremental strategies of the two actions a₆ and a₇ at node 5 can be denoted as f^(t+1)(a₆|I⁵) and f^(t+1)(a₇|I⁵) as the incremental strategies can be used for determining strategies in the next iteration (e.g., (t+1)-th iteration). The incremental strategies can represent probabilities of each of the two actions a₆ and a₇ at node 5 that lead to node 6 and node 7, respectively.

The FCFV of node 5, {hacek over (v)}(I⁵), can be calculated based on the incremental strategy f^(t), for example, according to {hacek over (v)}(I⁵)=Σ_(a)v(a|I⁵)f(a|I⁵)=v(a₆|I⁵)f^(t)(a₆|I⁵)+v(a₇|I⁵)f^(t)(a₇|I⁵). The FCFV of node 5, {hacek over (v)}(I⁵), can be used to replace the CFV of the node 5, v(I⁵), that is, v^(t)(I⁵)={hacek over (v)}^(t)(I⁵), for example, for calculating the FCFV of a parent node of node 5, that is, node 1 as shown in 500 b.

For example, as shown in 500 b, node 1 has two actions (denoted as a₄ and a₅) leading to node 4 and node 5, respectively. Since node 4 is a leaf node, its CFV can be regarded as its FCFV. A regret value of the node 1 can be calculated based on the CFV of node 5, v(I⁵), which is updated to be the FCFV of node 5, {hacek over (v)}(I⁵). For example, the regret value of action a₄ in the state of the node 1 can be calculated according to r(a₄|I¹)=v(a₄|I¹)−v(I¹). Similarly, the regret value of action a₅ in the state of the node 1 can be calculated according to r(a₅|I¹)=v(a₅|I¹)−v(I¹).

Based on the regret values of the node 1, incremental strategies (denoted as f^(t) (a₄|I¹) and f^(t)(a₅|I¹)) of the two actions a₄ and a₅ at node 1 can be calculated, for example, according to the regret matching of the streamline CFR algorithm as shown in Eq. (8). The incremental strategies can represent probabilities of each of the two actions a₄ and a₅ at node 1 that lead to node 4 and node 5, respectively.

The FCFV of node 1, {hacek over (v)}(I¹), can be calculated based on the incremental strategy f^(t), for example, according to v(a₄|I¹)f^(t)(a₄|I₁)+v(a₅|I¹)f^(t)(a₅|I¹). The FCFV of node 1, {hacek over (v)}(I¹), can be used to replace the CFV of the node 1, v(I¹), for example, for calculating the CFV of a parent node of node 1, that is, node 0 as shown in 500 b.

The above bottom-up process for calculating the FCFVs of nodes of the game tree can be continued until the root node is reached. In some embodiments, the FCFVs of nodes can be used in place of their respective CFVs for determining action selection policies (e.g., strategies), for example, by performing an original CFR, MCCFR, streamline CFR, or any other variations of CFR algorithms. As an example, the FCFVs of nodes of the game tree can be used in connection with the a streamline CFR where the strategy in the next iteration can be updated based on the incremental strategy f^(t)(a|I_(i)) according to σ^(t+1)(a|I_(i))=(1−α)σ^(t)(a|I_(i))+αf^(t+1)(a|I_(i)), wherein α is a learning rate that approaches 0 as t approaches infinity.

FIG. 6 is a flowchart of an example of a process for performing a fictitious counterfactual regret minimization (CFR) for determining action selection policies for software applications in accordance with embodiments of this specification. The process 600 can be an example of the fictitious CFR algorithm described above with respect to FIG. 5.

In some embodiments, the process 600 can be performed in an iterative manner, for example, by performing two or more iterations. In some embodiments, the process 600 can be used in automatic control, robotics, or any other applications that involve action selections. In some embodiments, the process 600 can be performed by an execution device for generating an action selection policy (e.g., a strategy) for completing a task (e.g., finding Nash equilibrium) in an environment that includes the execution device and one or more other devices. In some embodiments, the execution device can perform the process 600 in for controlling operations of the execution device according to the action selection policy.

In some embodiments, the execution device can include a data processing apparatus such as a system of one or more computers, located in one or more locations, and programmed appropriately in accordance with this specification. For example, a computer system 700 of FIG. 7, appropriately programmed, can perform the process 600. The execution device can be associated with an execution party or player. The execution party or player and one or more other parties (e.g., associated with the one or more other devices) can be participants or players in an environment, for example, for strategy searching in strategic interaction between the execution party and one or more other parties.

In some embodiments, the environment can be modeled by an imperfect information game (IIG) that involves two or more players. In some embodiments, the process 600 can be performed for solving an IIG, for example, by the execution party supported by the application. The IIG can represent one or more real-world scenarios such as resource allocation, product/service recommendation, cyber-attack prediction and/or prevention, traffic routing, fraud management, etc., that involve two or more parties, where each party may have incomplete or imperfect information about the other party's decisions. As an example, the IIG can represent a collaborative product-service recommendation service that involves at least a first player and a second player. The first player may be, for example, an online retailer that has customer (or user) information, product and service information, purchase history of the customers, etc. The second player can be, for example, a social network platform that has social networking data of the customers, a bank or another finical institution that has financial information of the customers, a car dealership, or any other parties that may have information of the customers on the customers' preferences, needs, financial situations, locations, etc. in predicting and recommendations of products and services to the customers. The first player and the second player may each have proprietary data that the player does not want to share with others. The second player may only provide partial information to the first player at different times. As such, the first player may only have limited access to information of the second player. In some embodiments, the process 600 can be performed for making a recommendation to a party with limited information of the second party, planning a route with limited information.

At 602, an action selection policy (e.g., a strategy σ_(i) ^(t)) in a first iteration, i.e., t=1 iteration, is initialized. In some embodiments, an action selection policy can include or otherwise specify a respective probability (e.g., σ_(i) ^(t)(a_(j)|I_(i)) of selecting an action (e.g., a_(j)) among a plurality of possible actions in a current state (e.g., state i) of the execution device (e.g., the device of the execution device that perform the process 600). The current state results from a previous action taken by the execution device in a previous state, and each action of the plurality of possible actions leads to a respective next state if performed by the execution device when the execution device is in the current state. In some embodiments, a state can be represented by a node (e.g., node 0-7 as shown in 500 b with an corresponding information set) of the game tree that represents the environment and a plurality of possible actions in a state (e.g., node 5 as shown in 500 b) can include the multiple actions (e.g., actions a₆ and a₇ as shown in 500 b) of the state that leads to respective next states (e.g., node 6 and node 7 as shown in 500 b). As shown in 500 b, a state of the execution device (e.g., node 5) results from a previous action a₅ taken by the execution device in a previous state (e.g., node 1) and each action of the plurality of possible actions (e.g., actions a₆ and a₇) leads to a respective next state (e.g., nodes 6 and 7) if performed by the execution device when the execution device is in the current state (e.g., node 5).

In some embodiments, the strategy can be initialized, for example, based on an existing strategy, a uniform random strategy (e.g. a strategy based on a uniform probability distribution), or another strategy (e.g. a strategy based on a different probability distribution). For example, if the system warm starts from an existing CFR method (e.g., an original CFR or MCCFR method), the iterative strategy can be initialized from an existing strategy profile to clone existing regrets and strategy.

At 604, whether a convergence condition is met is determined. The convergence condition can be used for determining whether to continue or terminate the iteration. In some embodiments, the convergence condition can be based on exploitability of a strategy σ. According to the definition of exploitability, exploitability should be large than or equal to 0. The smaller exploitability indicates a better strategy. That is, the exploitability of converged strategy should approach 0 after enough iterations. For example, in poker, when the exploitability is less than 1, the time-average strategy is regarded as a good strategy and it is determined that the convergence condition is met. In some embodiments, the convergence condition can be based on a predetermined number of iterations. For example, in a small game, the iterations can be easily determined by the exploitability. That is, if exploitability is small enough, the process 600 can terminate. In a large game, the exploitability is intractable, typically a large parameter for iteration can be specified. After each iteration, a new strategy profile can be obtained, which is better than the old one. For example, in a large game, the process 600 can terminate after a sufficient number of iterations.

If the convergence condition is met, no further iteration is needed. The process 600 proceeds to 622, where the action selection policy in the current iteration (e.g., the strategy σ_(i) ^(t)) is outputted, for example, to control operations of the execution device according to the action selection policy. If the convergence condition is not met, t is increased by 1, and the process 600 proceeds to a next iteration, wherein t>1.

In a current iteration (e.g., t-th iteration), for each action among a plurality of possible actions in a state of the execution device, at 605, an action selection policy in the current iteration (e.g., the strategy σ_(i) ^(t)) is obtained.

At 606, a respective first reward for each action in the current state is obtained. In some embodiments, the respective first reward for each action represents a gain attributed to the action towards completing the task. For example, the first reward for each action can be a CFV of each action. In some embodiments, obtaining a respective first reward of the each action comprises obtaining a respective first reward of the each action by traversing a game tree that represents the environment based on an action selection policy in a previous iteration (e.g., the strategy σ_(i) ^(t−1)).

In some embodiments, each iteration of the process 600 can include a bottom-up process for updating first rewards of the states. For example, the process 600 can start from terminal states (e.g., the leaf node 6 and node 7 as shown in 500 b) and move up to the initial state (e.g., the root node 0 as shown in 500 b). In some embodiments, for terminal nodes, a respective first reward of the each action in the terminal state can be the first reward of the terminal state (e.g., utility function u_(i)(z) or a payoff of a terminal state z) because the terminal state has no further action leading to any next state.

At 608, a first reward of the current state is computed based on the respective first rewards for the actions and the action selection policy in the current iteration. In some embodiments, the first reward for the current state represents a gain attributed to the current state towards completing the task. For example, the first reward for the current state can be a CFV of the current state.

In some embodiments, computing a first reward of the current state (e.g., a non-terminal state) based on the respective first rewards for the actions and the action selection policy in the current iteration comprises computing the first reward of the current state based on a sum of the respective first rewards for actions weighted by corresponding probabilities of selecting the actions in the current iteration, for example, according to Eq. (11).

As an example, as shown in 500 b, the first reward for a current state (e.g., the CFV of node 5, v(I⁵)) can be calculated based on the first reward of action a₆ (e.g., the CFV of action a₆, v(a₆|I⁵)), the first reward of action a₇ (e.g., the CFV of the action a₇, v(a₇|I⁵) and the action selection policy in the current iteration σ^(t) such as the probabilities of selecting the actions in the current iteration, for example, according to v(I⁵)=Σ_(a)v(a|I⁵)σ(a|I⁵)=v(a₆|I⁵)σ^(t)(a₆|I⁵)+v(a₇|I⁵)σ^(t)(a₇|I⁵).

At 610, a respective regret value for each action of the plurality of possible actions is computed based on a difference between the respective first reward for the action and the first reward for the current state. In some embodiments, the regret value of the action in the state of the execution device represents a difference between a gain or utility of the execution device after taking the action in the state and a gain or utility of the execution device in the state (without taking the action). As an example, as shown in 500 b, the regret value of action a₆ in the state of the node 5 can be calculated according to r(a₆|I⁵)=v(a₆|I⁵)−v(I⁵). Similarly, the regret values of action a₇ in the state of the node 5 can be calculated according to r(a₇|I⁵)=v(a₆|I⁵)−v(I⁵).

At 612, an incremental action selection policy is computed based on the respective regret value of the each action in the current iteration. The incremental action selection policy can be used for determining an action selection policy in a next iteration (e.g., σ_(i) ^(t+1)), for example, according to Eq. (9). The incremental action selection policy can include or otherwise specify a respective probability of the each action (e.g. σ̆_(i) ^(t+1)(a|I_(i)) or f^(t+1)(a|I_(i))). In some embodiments, the incremental action selection policy is computed based on the respective regret value of the each action in the current iteration but not any regret value of the each action in any iteration prior to the current iteration, for example, according to Eq. (8).

At 614, a second reward for the current state is computed based on the respective first rewards for the actions and the incremental action selection policy, wherein the second reward for the state includes partial information of the action selection policy in the next iteration. In some embodiments, the second reward of the state of the execution device is computed based on a weighted sum of the respective first reward for the each action and the respective probability of the each action. In some embodiments, the second reward of the state of the execution device can be the FCFV of the state of the execution device. In some embodiments, the second reward of the state of the execution device can be computed according to Eq. (10). For example, as shown in 500 b, the second reward of the state of the execution device (e.g., the FCFV of node 5, {hacek over (v)}(I⁵) can be calculated based on a weighted sum of the respective first reward for the each action (e.g., the CFV of action a₆, v(a₆|I⁵) and the CFV of the action a₇, v(a₇|I⁵)) and the respective probability of the each action (e.g., the incremental strategy f^(t)), for example, according to {hacek over (v)}(I⁵)=Σ_(a)v(a|I⁵)f(a|I⁵)=v(a₆|I⁵)f^(t)(a₆|I⁵)+v(a₇|I⁵)f^(t)(a₇|I⁵).

At 616, the first reward for the current state is replaced with the second reward for the current state. In some embodiments, the first reward of the state is replaced with the second reward of the state to represent a first reward for the previous action taken by the execution device in the previous state for updating the first reward of the previous state based on the second reward of the state. For example, as shown in 500 b, the first reward of the state (e.g., the CFV of the node 5, v(I⁵)) is updated to be the second reward of the state (e.g., the FCFV of node 5, {hacek over (v)}(I⁵)) to represent a first reward for the previous action taken by the execution device in the previous state (e.g., the CFV of action a₅ in the previous state of node 1, v(a₅|I¹)), for example, for updating the first reward of the previous state (e.g., the CFV of the node 1, v(I¹)) based on the second reward of the state (e.g., the FCFV of node 5, {hacek over (v)}(I⁵)).

In some embodiments, replacing the first reward for the current state with the second reward for the current state can simply the algorithm and improve the storage efficiency as no additional storage space needs to be allocated to store the second reward of the state.

At 618, whether the current state is the initial state is determined. In some embodiments, such a determination can be used for determining whether to continue or terminate updating the first rewards of the states in the current iteration. In some embodiments, the initial state can be represented by a root node of the game tree (e.g., node 0 as shown in 500 b).

If the current state is the initial state, no further updating of the first reward is needed. The process 600 proceeds to 620. If the current state is not the initial state, a previous state of the state (e.g., a parent node of the current node such as node 1 of the current node 5 as shown in 500 b) is used to replace the current state, and the process 600 goes back to 606 to obtain a respective first reward for each action (e.g., action a₄ and action a₅ as shown in 500 b) of the previous state (e.g., node 1 as shown in 500 b). The process 600 proceeds as shown in FIG. 6, for example, by computing a first reward of the previous state based on the respective first reward of the each action in the previous state at 608, computing a respective regret value of the each action in the previous state based on a difference between the respective first reward of the each action in the previous state and the first reward of the previous state at 610, computing a respective probability of the each action in the previous state based on the respective regret value of the each action in the previous state at 612, computing a second reward of the previous state of the execution device based on a weighted sum of the respective first reward of the each action in the previous state and the respective probability of the each action in the previous state at 614; and updating the first reward of the previous state to be the second reward of the previous state at 616.

At 620, the action selection policy in the next iteration (e.g., σ_(i) ^(t+1)) is determined based on the second reward of the current state. In some embodiments, determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on the action selection policy in the current iteration (e.g., σ_(i) ^(t)) and the incremental action selection policy (e.g., σ̆_(i) ^(t+1)(a|I_(i)) or f^(t+1)(a|I_(i))) but not on any regret value of the action in the state of the device in any iteration prior to the current iteration. For example, determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on a weighted sum of the action selection policy in the current iteration (e.g., σ_(i) ^(t)) and the incremental action selection policy (e.g., σ̆_(i) ^(t+1)(a|I_(i)) or f^(t+1)(a|I_(i))), for example, according to Eq. (9).

In some embodiments, as discussed with respect to 500 b, determining the action selection policy in the next iteration based on the second reward of the current state (e.g., the FCFV of node 5, {hacek over (v)}(I⁵)) comprises computing a probability (e.g., σ̆_(i) ^(t+1)(a₅|I¹)) of selecting the previous action (e.g., a₅) among the plurality of possible actions in the previous state (e.g., node 1) in the next iteration based on the second reward for the current state (e.g., the FCFV of node 5, {hacek over (v)}(I⁵)). For example, σ̆_(i) ^(t+1)(a₅|I¹⁾⁾⁼⁽1−α^(t)(I¹))σ_(i) ^(t)(a₅|I¹)+α^(t)(I¹)σ̆_(i) ^(t+1)(a₅|I¹), wherein σ̆_(i) ^(t+1)(a₅|I¹) (i.e., the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration) can be computed by: computing a first reward for the previous state (e.g., the CFV of the node 1) based on the second reward for the current state (e.g., the FCFV of node 5, {hacek over (V)}(I⁵)); computing a regret value for the previous action (e.g., r(a₅|I⁴)) based on a difference between the first reward for the previous action in the previous state and the first reward for the previous state; and computing the σ̆_(i) ^(t+1)(a₅|I¹) (i.e., the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration) based on the regret value of the previous action (e.g., r(a₅|I⁴)), for example, according to Eq. (8).

At 622, actions of the execution device are controlled according to the action selection policy. For example, the action selection policy can serve as an output of the software-implemented application to automatically control the execution device's action at each state, for example, by selecting the action that has the highest probability among a plurality of possible actions based on the action selection policy. As an example, the environment comprises a traffic routing environment, the execution device supported by the application comprises a computer-assisted vehicle, the action selection policy comprises a route selection policy for controlling directions of the computer-assisted vehicle, and controlling operations of the execution device according to the action selection policy comprises controlling directions of the computer-assisted vehicle according to the route selection policy.

FIG. 7 depicts a block diagram illustrating an example of a computer-implemented system 700 used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures in accordance with embodiments of this specification. In the illustrated embodiment, System 700 includes a Computer 702 and a Network 730.

The illustrated Computer 702 is intended to encompass any computing device such as a server, desktop computer, laptop/notebook computer, wireless data port, smart phone, personal data assistant (PDA), tablet computer, one or more processors within these devices, another computing device, or a combination of computing devices, including physical or virtual instances of the computing device, or a combination of physical or virtual instances of the computing device. Additionally, the Computer 702 can include an input device, such as a keypad, keyboard, touch screen, another input device, or a combination of input devices that can accept user information, and an output device that conveys information associated with the operation of the Computer 702, including digital data, visual, audio, another type of information, or a combination of types of information, on a graphical-type user interface (UI) (or GUI) or other UI.

The Computer 702 can serve in a role in a distributed computing system as a client, network component, a server, a database or another persistency, another role, or a combination of roles for performing the subject matter described in the present disclosure. The illustrated Computer 702 is communicably coupled with a Network 730. In some embodiments, one or more components of the Computer 702 can be configured to operate within an environment, including cloud-computing-based, local, global, another environment, or a combination of environments.

At a high level, the Computer 702 is an electronic computing device operable to receive, transmit, process, store, or manage data and information associated with the described subject matter. According to some embodiments, the Computer 702 can also include or be communicably coupled with a server, including an application server, e-mail server, web server, caching server, streaming data server, another server, or a combination of servers.

The Computer 702 can receive requests over Network 730 (for example, from a client software application executing on another Computer 702) and respond to the received requests by processing the received requests using a software application or a combination of software applications. In addition, requests can also be sent to the Computer 702 from internal users (for example, from a command console or by another internal access method), external or third-parties, or other entities, individuals, systems, or computers.

Each of the components of the Computer 702 can communicate using a System Bus 703. In some embodiments, any or all of the components of the Computer 702, including hardware, software, or a combination of hardware and software, can interface over the System Bus 703 using an application programming interface (API) 712, a Service Layer 713, or a combination of the API 712 and Service Layer 713. The API 712 can include specifications for routines, data structures, and object classes. The API 712 can be either computer-language independent or dependent and refer to a complete interface, a single function, or even a set of APIs. The Service Layer 713 provides software services to the Computer 702 or other components (whether illustrated or not) that are communicably coupled to the Computer 702. The functionality of the Computer 702 can be accessible for all service consumers using the Service Layer 713. Software services, such as those provided by the Service Layer 713, provide reusable, defined functionalities through a defined interface. For example, the interface can be software written in JAVA, C++, another computing language, or a combination of computing languages providing data in extensible markup language (XML) format, another format, or a combination of formats. While illustrated as an integrated component of the Computer 702, alternative embodiments can illustrate the API 712 or the Service Layer 713 as stand-alone components in relation to other components of the Computer 702 or other components (whether illustrated or not) that are communicably coupled to the Computer 702. Moreover, any or all parts of the API 712 or the Service Layer 713 can be implemented as a child or a sub-module of another software module, enterprise application, or hardware module without departing from the scope of the present disclosure.

The Computer 702 includes an Interface 704. Although illustrated as a single Interface 704, two or more Interfaces 704 can be used according to particular needs, desires, or particular embodiments of the Computer 702. The Interface 704 is used by the Computer 702 for communicating with another computing system (whether illustrated or not) that is communicatively linked to the Network 730 in a distributed environment. Generally, the Interface 704 is operable to communicate with the Network 730 and includes logic encoded in software, hardware, or a combination of software and hardware. More specifically, the Interface 704 can include software supporting one or more communication protocols associated with communications such that the Network 730 or hardware of Interface 704 is operable to communicate physical signals within and outside of the illustrated Computer 702.

The Computer 702 includes a Processor 705. Although illustrated as a single Processor 705, two or more Processors 705 can be used according to particular needs, desires, or particular embodiments of the Computer 702. Generally, the Processor 705 executes instructions and manipulates data to perform the operations of the Computer 702 and any algorithms, methods, functions, processes, flows, and procedures as described in the present disclosure.

The Computer 702 also includes a Database 706 that can hold data for the Computer 702, another component communicatively linked to the Network 730 (whether illustrated or not), or a combination of the Computer 702 and another component. For example, Database 706 can be an in-memory, conventional, or another type of database storing data consistent with the present disclosure. In some embodiments, Database 706 can be a combination of two or more different database types (for example, a hybrid in-memory and conventional database) according to particular needs, desires, or particular embodiments of the Computer 702 and the described functionality. Although illustrated as a single Database 706, two or more databases of similar or differing types can be used according to particular needs, desires, or particular embodiments of the Computer 702 and the described functionality. While Database 706 is illustrated as an integral component of the Computer 702, in alternative embodiments, Database 706 can be external to the Computer 702. As an example, Database 706 can include the above-described strategies 716 of a CFR algorithm.

The Computer 702 also includes a Memory 707 that can hold data for the Computer 702, another component or components communicatively linked to the Network 730 (whether illustrated or not), or a combination of the Computer 702 and another component. Memory 707 can store any data consistent with the present disclosure. In some embodiments, Memory 707 can be a combination of two or more different types of memory (for example, a combination of semiconductor and magnetic storage) according to particular needs, desires, or particular embodiments of the Computer 702 and the described functionality. Although illustrated as a single Memory 707, two or more Memories 707 or similar or differing types can be used according to particular needs, desires, or particular embodiments of the Computer 702 and the described functionality. While Memory 707 is illustrated as an integral component of the Computer 702, in alternative embodiments, Memory 707 can be external to the Computer 702.

The Application 708 is an algorithmic software engine providing functionality according to particular needs, desires, or particular embodiments of the Computer 702, particularly with respect to functionality described in the present disclosure. For example, Application 708 can serve as one or more components, modules, or applications. Further, although illustrated as a single Application 708, the Application 708 can be implemented as multiple Applications 708 on the Computer 702. In addition, although illustrated as integral to the Computer 702, in alternative embodiments, the Application 708 can be external to the Computer 702.

The Computer 702 can also include a Power Supply 714. The Power Supply 714 can include a rechargeable or non-rechargeable battery that can be configured to be either user- or non-user-replaceable. In some embodiments, the Power Supply 714 can include power-conversion or management circuits (including recharging, standby, or another power management functionality). In some embodiments, the Power Supply 714 can include a power plug to allow the Computer 702 to be plugged into a wall socket or another power source to, for example, power the Computer 702 or recharge a rechargeable battery.

There can be any number of Computers 702 associated with, or external to, a computer system containing Computer 702, each Computer 702 communicating over Network 730. Further, the term “client,” “user,” or other appropriate terminology can be used interchangeably, as appropriate, without departing from the scope of the present disclosure. Moreover, the present disclosure contemplates that many users can use one Computer 702, or that one user can use multiple computers 702.

FIG. 8 is a diagram of an example of modules of an apparatus 800 in accordance with embodiments of this specification. In some embodiments, the apparatus 800 can perform a computer-implemented method for an execution device for generating an action selection policy for completing a task in an environment that includes the execution device and one or more other devices. In some embodiments, the method represents the environment, possible actions of parties, and imperfect information available to the application about the other parties with data representing an imperfect information game (IIG), wherein the application determines the actionable output by performing a counterfactual regret minimization (CFR) for strategy searching in strategic interaction between the parties in an iterative manner, for example, by performing two or more iterations.

The apparatus 800 can correspond to the embodiments described above, and the apparatus 800 includes the following: a first obtaining module 801 for obtaining an action selection policy in a current iteration of a plurality of iterations, wherein the action selection policy specifies a respective probability of selecting an action among a plurality of possible actions in a current state of the execution device, wherein the current state results from a previous action taken by the execution device in a previous state, and each action of the plurality of possible actions leads to a respective next state if performed by the execution device when the execution device is in the current state; a second obtaining module 802 for obtaining a respective first reward for each action in the current state, wherein the respective first reward for each action represents a gain attributed to the action towards completing the task; a first computing module 803 for computing a first reward of the current state based on the respective first rewards for the actions and the action selection policy in the current iteration, wherein the first reward for the current state represents a gain attributed to the current state towards completing the task; a second computing module 804 for computing a respective regret value of each action of the plurality of possible actions based on a difference between the respective first reward for the action and the first reward for the current state; a third computing module 805 for computing an incremental action selection policy based on the respective regret value of the each action in the current iteration but not any regret value of the each action in any iteration prior to the current iteration, wherein the incremental action selection policy is used for determining an action selection policy in a next iteration; a fourth computing module 806 for computing a second reward for the current state based on the respective first rewards for the actions and the incremental action selection policy, wherein the second reward for the state includes partial information of the action selection policy in the next iteration; a determining module 808 for determining the action selection policy in the next iteration based on the second reward of the current state; and a controlling module 809 for controlling actions of the execution device according to the action selection policy.

In an optional embodiment, wherein obtaining a respective first reward of the each action comprises obtaining a respective first reward of the each action by traversing a game tree that represents the environment based on an action selection policy in a previous iteration.

In an optional embodiment, wherein computing a first reward of the current state based on the respective first rewards for the actions and the action selection policy in the current iteration comprises computing the first reward of the current state based on a sum of the respective first rewards for actions weighted by corresponding probabilities of selecting the actions in the current iteration.

In an optional embodiment, wherein determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on a weighted sum of the action selection policy in the current iteration and the incremental action selection policy.

In an optional embodiment, the apparatus 800 further comprising a replacing module 807 for replacing the first reward for the current state with the second reward for the current state.

In an optional embodiment, wherein determining the action selection policy in the next iteration based on the second reward of the current state comprises computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state.

In an optional embodiment, wherein computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state comprises: computing a first reward for the previous state based on the second reward for the current state; computing a regret value of the previous action based on a difference between the first reward for the previous action in the previous state and the second reward for the previous state; and computing the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the regret value of the previous action.

In an optional embodiment, wherein: the environment comprises a traffic routing environment, the execution device supported by the application comprises a computer-assisted vehicle, the action selection policy comprises a route selection policy for controlling directions of the computer-assisted vehicle, and controlling operations of the execution device according to the action selection policy comprises controlling directions of the computer-assisted vehicle according to the route selection policy.

The system, apparatus, module, or unit illustrated in the previous embodiments can be implemented by using a computer chip or an entity, or can be implemented by using a product having a certain function. A typical embodiment device is a computer, and the computer can be a personal computer, a laptop computer, a cellular phone, a camera phone, a smartphone, a personal digital assistant, a media player, a navigation device, an email receiving and sending device, a game console, a tablet computer, a wearable device, or any combination of these devices.

For an embodiment process of functions and roles of each module in the apparatus, references can be made to an embodiment process of corresponding steps in the previous method. Details are omitted here for simplicity.

Because an apparatus embodiment basically corresponds to a method embodiment, for related parts, references can be made to related descriptions in the method embodiment. The previously described apparatus embodiment is merely an example. The modules described as separate parts may or may not be physically separate, and parts displayed as modules may or may not be physical modules, may be located in one position, or may be distributed on a number of network modules. Some or all of the modules can be selected based on actual demands to achieve the objectives of the solutions of the specification. A person of ordinary skill in the art can understand and implement the embodiments of the present application without creative efforts.

Referring again to FIG. 8, it can be interpreted as illustrating an internal functional module and a structure of a data processing apparatus for generating an action selection policy for a software-implemented application that performs actions in an environment that includes an execution party supported by the application and one or more other parties. An execution body in essence can be an electronic device, and the electronic device includes the following: one or more processors and a memory configured to store an executable instruction of the one or more processors.

The techniques described in this specification produce one or more technical effects. In some embodiments, the described techniques can be performed by an execution device for generating an action selection policy for completing a task in an environment that includes the execution device and one or more other devices. In some embodiments, the described techniques can determine an action selection policy for a software-implemented application that performs actions in an environment that includes an execution party supported by the application and one or more other parties. In some embodiments, the described techniques can be used in automatic control, robotics, or any other applications that involve action selections.

In some embodiments, the described techniques can help find better strategies of real-world scenarios such as resource allocation, product/service recommendation, cyber-attack prediction and/or prevention, traffic routing, fraud management, etc. that can be modeled or represented by strategic interaction between parties, such as, an IIG that involves two or more parties in a more efficient manner.

In some embodiments, the described techniques can improve the convergence speed of a counterfactual regret minimization (CFR) algorithm in finding Nash equilibrium for solving a game that represents one or more real-world scenarios. In some embodiments, the described techniques can improve computational efficiency and reduce the computational load of the CFR algorithm in finding the best strategies of the real-world scenarios modeled by the IIG, for example, by using an incremental strategy, rather than an accumulative regret or average strategy, in updating the strategy and regret values for each iteration of the CFR algorithm.

In some embodiments, the disclosed streamline CFR algorithm and the fictitious CFR algorithm can save memory space. For example, the disclosed streamline CFR algorithm and the fictitious CFR algorithm may need only half of the amount of memory space required by the existing CFR algorithm while converging to comparable results produced by the original CFR. The disclosed streamline CFR algorithm and the fictitious CFR algorithm can be used in large games even with memory constraints.

In some embodiments, the disclosed fictitious CFR algorithm can provide faster convergence compared to the original CFR algorithm. For example, the fictitious CFR algorithm can use FCFVs to take advantage of an updated incremental strategy computed based on CFVs in a current iteration without waiting until the next iteration. As such, the fictitious CFR algorithm can achieve a convergence faster than the original CFR algorithm.

Described embodiments of the subject matter can include one or more features, alone or in combination.

For example, in a first embodiment, a computer-implemented method of an execution device for generating an action selection policy for completing a task in an environment that includes the execution device and one or more other devices. The method includes, in a current iteration of a plurality of iterations, obtaining an action selection policy in the current iteration, wherein the action selection policy specifies a respective probability of selecting an action among a plurality of possible actions in a current state of the execution device, wherein the current state results from a previous action taken by the execution device in a previous state, and each action of the plurality of possible actions leads to a respective next state if performed by the execution device when the execution device is in the current state; obtaining a respective first reward for each action in the current state, wherein the respective first reward for each action represents a gain attributed to the action towards completing the task; computing a first reward of the current state based on the respective first rewards for the actions and the action selection policy in the current iteration, wherein the first reward for the current state represents a gain attributed to the current state towards completing the task; computing a respective regret value of each action of the plurality of possible actions based on a difference between the respective first reward for the action and the first reward for the current state; computing an incremental action selection policy based on the respective regret value of the each action in the current iteration but not any regret value of the each action in any iteration prior to the current iteration, wherein the incremental action selection policy is used for determining an action selection policy in a next iteration; computing a second reward for the current state based on the respective first rewards for the actions and the incremental action selection policy, wherein the second reward for the state includes partial information of the action selection policy in the next iteration; determining the action selection policy in the next iteration based on the second reward of the current state; and controlling actions of the execution device according to the action selection policy.

The foregoing and other described embodiments can each, optionally, include one or more of the following features:

A first feature, combinable with any of the following features, wherein obtaining a respective first reward of the each action comprises obtaining a respective first reward of the each action by traversing a game tree that represents the environment based on an action selection policy in a previous iteration.

A second feature, combinable with any of the following features, wherein computing a first reward of the current state based on the respective first rewards for the actions and the action selection policy in the current iteration comprises computing the first reward of the current state based on a sum of the respective first rewards for actions weighted by corresponding probabilities of selecting the actions in the current iteration.

A third feature, combinable with any of the following features, wherein determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on a weighted sum of the action selection policy in the current iteration and the incremental action selection policy.

A fourth feature, combinable with any of the following features, further comprising replacing the first reward for the current state with the second reward for the current state.

A fifth feature, combinable with any of the following features, wherein determining the action selection policy in the next iteration based on the second reward of the current state comprises computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state.

A sixth feature, combinable with any of the following features, wherein computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state comprises: computing a first reward for the previous state based on the second reward for the current state; computing a regret value of the previous action based on a difference between the first reward for the previous action in the previous state and the second reward for the previous state; and computing the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the regret value of the previous action.

A seventh feature, combinable with any of the following features, wherein: the environment comprises a traffic routing environment, the execution device supported by the application comprises a computer-assisted vehicle, the action selection policy comprises a route selection policy for controlling directions of the computer-assisted vehicle, and controlling operations of the execution device according to the action selection policy comprises controlling directions of the computer-assisted vehicle according to the route selection policy.

Embodiments of the subject matter and the actions and operations described in this specification can be implemented in digital electronic circuitry, in tangibly-embodied computer software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Embodiments of the subject matter described in this specification can be implemented as one or more computer programs, e.g., one or more modules of computer program instructions, encoded on a computer program carrier, for execution by, or to control the operation of, data processing apparatus. For example, a computer program carrier can include one or more computer-readable storage media that have instructions encoded or stored thereon. The carrier may be a tangible non-transitory computer-readable medium, such as a magnetic, magneto optical, or optical disk, a solid state drive, a random access memory (RAM), a read-only memory (ROM), or other types of media. Alternatively, or in addition, the carrier may be an artificially generated propagated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal that is generated to encode information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The computer storage medium can be or be part of a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of them. A computer storage medium is not a propagated signal.

A computer program, which may also be referred to or described as a program, software, a software application, an app, a module, a software module, an engine, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages; and it can be deployed in any form, including as a stand-alone program or as a module, component, engine, subroutine, or other unit suitable for executing in a computing environment, which environment may include one or more computers interconnected by a data communication network in one or more locations.

A computer program may, but need not, correspond to a file in a file system. A computer program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub programs, or portions of code.

Processors for execution of a computer program include, by way of example, both general- and special-purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive the instructions of the computer program for execution as well as data from a non-transitory computer-readable medium coupled to the processor.

The term “data processing apparatus” encompasses all kinds of apparatuses, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. Data processing apparatus can include special-purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application specific integrated circuit), or a GPU (graphics processing unit). The apparatus can also include, in addition to hardware, code that creates an execution environment for computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

The processes and logic flows described in this specification can be performed by one or more computers or processors executing one or more computer programs to perform operations by operating on input data and generating output. The processes and logic flows can also be performed by special-purpose logic circuitry, e.g., an FPGA, an ASIC, or a GPU, or by a combination of special-purpose logic circuitry and one or more programmed computers.

Computers suitable for the execution of a computer program can be based on general or special-purpose microprocessors or both, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read only memory or a random access memory or both. Elements of a computer can include a central processing unit for executing instructions and one or more memory devices for storing instructions and data. The central processing unit and the memory can be supplemented by, or incorporated in, special-purpose logic circuitry.

Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to one or more storage devices. The storage devices can be, for example, magnetic, magneto optical, or optical disks, solid state drives, or any other type of non-transitory, computer-readable media. However, a computer need not have such devices. Thus, a computer may be coupled to one or more storage devices, such as, one or more memories, that are local and/or remote. For example, a computer can include one or more local memories that are integral components of the computer, or the computer can be coupled to one or more remote memories that are in a cloud network. Moreover, a computer can be embedded in another device, e.g., a mobile telephone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a Global Positioning System (GPS) receiver, or a portable storage device, e.g., a universal serial bus (USB) flash drive, to name just a few.

Components can be “coupled to” each other by being commutatively such as electrically or optically connected to one another, either directly or via one or more intermediate components. Components can also be “coupled to” each other if one of the components is integrated into the other. For example, a storage component that is integrated into a processor (e.g., an L2 cache component) is “coupled to” the processor.

To provide for interaction with a user, embodiments of the subject matter described in this specification can be implemented on, or configured to communicate with, a computer having a display device, e.g., a LCD (liquid crystal display) monitor, for displaying information to the user, and an input device by which the user can provide input to the computer, e.g., a keyboard and a pointing device, e.g., a mouse, a trackball or touchpad. Other kinds of devices can be used to provide for interaction with a user as well; for example, feedback provided to the user can be any form of sensory feedback, e.g., visual feedback, auditory feedback, or tactile feedback; and input from the user can be received in any form, including acoustic, speech, or tactile input. In addition, a computer can interact with a user by sending documents to and receiving documents from a device that is used by the user; for example, by sending web pages to a web browser on a user's device in response to requests received from the web browser, or by interacting with an app running on a user device, e.g., a smartphone or electronic tablet. Also, a computer can interact with a user by sending text messages or other forms of message to a personal device, e.g., a smartphone that is running a messaging application, and receiving responsive messages from the user in return.

This specification uses the term “configured to” in connection with systems, apparatus, and computer program components. For a system of one or more computers to be configured to perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For special-purpose logic circuitry to be configured to perform particular operations or actions means that the circuitry has electronic logic that performs the operations or actions.

While this specification contains many specific embodiment details, these should not be construed as limitations on the scope of what is being claimed, which is defined by the claims themselves, but rather as descriptions of features that may be specific to particular embodiments. Certain features that are described in this specification in the context of separate embodiments can also be realized in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiments can also be realized in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially be claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claim may be directed to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings and recited in the claims in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the embodiments described above should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

Particular embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. 

What is claimed is:
 1. A computer-implemented method for generating an action selection policy for causing an execution device to complete a task in an environment that includes the execution device and one or more other devices, the method comprising: at a current iteration of a plurality of iterations, obtaining, by a computer system, an action selection policy in the current iteration, wherein the action selection policy specifies a respective probability of selecting an action among a plurality of possible actions in a current state of the execution device, wherein the current state results from a previous action taken by the execution device in a previous state, and each action of the plurality of possible actions leads to a respective next state if performed by the execution device when the execution device is in the current state; obtaining, by the computer system, a respective first reward for each action of the plurality of possible actions in the current state, wherein the respective first reward for the action represents a gain attributed to the action towards completing the task; computing, by the computer system, a first reward for the current state based on respective first rewards for the plurality of possible actions and the action selection policy in the current iteration, wherein the first reward for the current state represents a gain attributed to the current state towards completing the task; computing, by the computer system, a respective regret value of each action of the plurality of possible actions based on a difference between the respective first reward for the action and the first reward for the current state; computing, by the computer system, an incremental action selection policy based on the respective regret value of the action in the current iteration but not any regret value of the action in any iteration prior to the current iteration, wherein the incremental action selection policy is used for determining an action selection policy in a next iteration; computing, by the computer system, a second reward for the current state based on the respective first rewards for the plurality of possible actions and the incremental action selection policy, wherein the second reward for the current state includes partial information of the action selection policy in the next iteration; and determining, by the computer system, the action selection policy in the next iteration based on the second reward of the current state; and controlling actions of the execution device according to the action selection policy.
 2. The method of claim 1, wherein obtaining a respective first reward for each action comprises obtaining a respective first reward for the action by traversing a game tree that represents the environment based on an action selection policy in a previous iteration.
 3. The method of claim 1, wherein computing a first reward for the current state based on respective first rewards for the plurality of possible actions and the action selection policy in the current iteration comprises computing the first reward for the current state based on a sum of the respective first rewards for the plurality of possible actions weighted by corresponding probabilities of selecting the plurality of possible actions in the current iteration.
 4. The method of claim 1, wherein determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on a weighted sum of the action selection policy in the current iteration and the incremental action selection policy.
 5. The method of claim 1, further comprising replacing the first reward for the current state with the second reward for the current state.
 6. The method of claim 1, wherein determining the action selection policy in the next iteration based on the second reward of the current state comprises computing a probability of selecting the previous action among a plurality of possible actions in the previous state in the next iteration based on the second reward for the current state.
 7. The method of claim 6, wherein computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state comprises: computing a first reward for the previous state based on the second reward for the current state; computing a regret value of the previous action based on a difference between the first reward for the previous action in the previous state and the first reward for the previous state; and computing the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the regret value of the previous action.
 8. The method of claim 1, wherein: the environment comprises a traffic routing environment, the execution device comprises a computer-assisted vehicle, the action selection policy comprises a route selection policy for controlling directions of the computer-assisted vehicle, and controlling operations of the execution device according to the action selection policy comprises controlling directions of the computer-assisted vehicle according to the route selection policy.
 9. A system for generating an action selection policy for completing a task in an environment that includes an execution device and one or more other devices, the system comprising: one or more processors; and one or more computer-readable memories coupled to the one or more processors and having instructions stored thereon that are executable by the one or more processors to perform operations comprising: at a current iteration of a plurality of iterations, obtaining an action selection policy in the current iteration, wherein the action selection policy specifies a respective probability of selecting an action among a plurality of possible actions in a current state of the execution device, wherein the current state results from a previous action taken by the execution device in a previous state, and each action of the plurality of possible actions leads to a respective next state if performed by the execution device when the execution device is in the current state; obtaining a respective first reward for each action of the plurality of possible actions in the current state, wherein the respective first reward for the action represents a gain attributed to the action towards completing the task; computing a first reward for the current state based on respective first rewards for the plurality of possible actions and the action selection policy in the current iteration, wherein the first reward for the current state represents a gain attributed to the current state towards completing the task; computing a respective regret value of each action of the plurality of possible actions based on a difference between the respective first reward for the action and the first reward for the current state; computing an incremental action selection policy based on the respective regret value of the action in the current iteration but not any regret value of the action in any iteration prior to the current iteration, wherein the incremental action selection policy is used for determining an action selection policy in a next iteration; computing a second reward for the current state based on the respective first rewards for the plurality of possible actions and the incremental action selection policy, wherein the second reward for the current state includes partial information of the action selection policy in the next iteration; and determining the action selection policy in the next iteration based on the second reward of the current state; and controlling actions of the execution device according to the action selection policy.
 10. The system of claim 9, wherein obtaining a respective first reward for each action comprises obtaining a respective first reward for the action by traversing a game tree that represents the environment based on an action selection policy in a previous iteration.
 11. The system of claim 9, wherein computing a first reward for the current state based on respective first rewards for the plurality of possible actions and the action selection policy in the current iteration comprises computing the first reward for the current state based on a sum of the respective first rewards for the plurality of possible actions weighted by corresponding probabilities of selecting the plurality of possible actions in the current iteration.
 12. The system of claim 9, wherein determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on a weighted sum of the action selection policy in the current iteration and the incremental action selection policy.
 13. The system of claim 9, the operations further comprising replacing the first reward for the current state with the second reward for the current state.
 14. The system of claim 9, wherein determining the action selection policy in the next iteration based on the second reward of the current state comprises computing a probability of selecting the previous action among a plurality of possible actions in the previous state in the next iteration based on the second reward for the current state.
 15. The system of claim 14, wherein computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state comprises: computing a first reward for the previous state based on the second reward for the current state; computing a regret value of the previous action based on a difference between the first reward for the previous action in the previous state and the first reward for the previous state; and computing the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the regret value of the previous action.
 16. The system of claim 9, wherein: the environment comprises a traffic routing environment, the execution device comprises a computer-assisted vehicle, the action selection policy comprises a route selection policy for controlling directions of the computer-assisted vehicle, and controlling operations of the execution device according to the action selection policy comprises controlling directions of the computer-assisted vehicle according to the route selection policy.
 17. A non-transitory, computer-readable storage medium storing one or more instructions executable by a computer system to perform operations for generating an action selection policy for completing a task in an environment that includes an execution device and one or more other devices, the operations comprising: at a current iteration of a plurality of iterations, obtaining an action selection policy in the current iteration, wherein the action selection policy specifies a respective probability of selecting an action among a plurality of possible actions in a current state of the execution device, wherein the current state results from a previous action taken by the execution device in a previous state, and each action of the plurality of possible actions leads to a respective next state if performed by the execution device when the execution device is in the current state; obtaining a respective first reward for each action of the plurality of possible actions in the current state, wherein the respective first reward for the action represents a gain attributed to the action towards completing the task; computing a first reward for the current state based on respective first rewards for the plurality of possible actions and the action selection policy in the current iteration, wherein the first reward for the current state represents a gain attributed to the current state towards completing the task; computing a respective regret value of each action of the plurality of possible actions based on a difference between the respective first reward for the action and the first reward for the current state; computing an incremental action selection policy based on the respective regret value of the action in the current iteration but not any regret value of the action in any iteration prior to the current iteration, wherein the incremental action selection policy is used for determining an action selection policy in a next iteration; computing a second reward for the current state based on the respective first rewards for the plurality of possible actions and the incremental action selection policy, wherein the second reward for the current state includes partial information of the action selection policy in the next iteration; and determining the action selection policy in the next iteration based on the second reward of the current state; and controlling actions of the execution device according to the action selection policy.
 18. The non-transitory, computer-readable storage medium of claim 17, wherein obtaining a respective first reward for each action comprises obtaining a respective first reward for the action by traversing a game tree that represents the environment based on an action selection policy in a previous iteration.
 19. The non-transitory, computer-readable storage medium of claim 17, wherein computing a first reward for the current state based on respective first rewards for the plurality of possible actions and the action selection policy in the current iteration comprises computing the first reward for the current state based on a sum of the respective first rewards for the plurality of possible actions weighted by corresponding probabilities of selecting the plurality of possible actions in the current iteration.
 20. The non-transitory, computer-readable storage medium of claim 17, wherein determining the action selection policy in the next iteration comprises determining the action selection policy in the next iteration based on a weighted sum of the action selection policy in the current iteration and the incremental action selection policy.
 21. The non-transitory, computer-readable storage medium of claim 17, the operations further comprising replacing the first reward for the current state with the second reward for the current state.
 22. The non-transitory, computer-readable storage medium of claim 17, wherein determining the action selection policy in the next iteration based on the second reward of the current state comprises computing a probability of selecting the previous action among a plurality of possible actions in the previous state in the next iteration based on the second reward for the current state.
 23. The non-transitory, computer-readable storage medium of claim 22, wherein computing a probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the second reward for the current state comprises: computing a first reward for the previous state based on the second reward for the current state; computing a regret value of the previous action based on a difference between the first reward for the previous action in the previous state and the first reward for the previous state; and computing the probability of selecting the previous action among the plurality of possible actions in the previous state in the next iteration based on the regret value of the previous action.
 24. The non-transitory, computer-readable storage medium of claim 17, wherein: the environment comprises a traffic routing environment, the execution device comprises a computer-assisted vehicle, the action selection policy comprises a route selection policy for controlling directions of the computer-assisted vehicle, and controlling operations of the execution device according to the action selection policy comprises controlling directions of the computer-assisted vehicle according to the route selection policy. 